The reactivity between lithium and the Al$_{13}$ cluster is studied using the density functional theory and the local density approximation (LDA) for exchange and correlation. The effects caused by the addition of two Li atoms on the structural and electronic properties of the Al cluster are analyzed by calculating equilibrium geometries, binding energies and ionization potentials. The corresponding positively and negatively charged species are also studied. The influence of the alkali atoms on the geometrical structure of the cluster is small, but the influence on the electronic structure is much larger. The cluster Al$_{13}$Li, with 40 valence electrons, has closed electronic shells and Al$_{13}$Li$_{2}$ has a single electron in the next shell. A striking result is the large drop of the ionization potential of the cluster induced by the addition of the second Li atom. These properties have been related to the changes of the electronic density by the analysis of density difference maps and atomic Hirshfeld charges. The photoabsorption spectrum has been calculated using the time-dependent version of the density functional theory.
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